Income Distribution, the Lorenz Curve and Tax Structures

Measuring inequality

The Lorenz curve plots the cumulative share of income (vertical axis) against the cumulative share of households ranked from poorest to richest (horizontal axis). Perfect equality would be the $45^{\circ}$ line: the bottom $20\%$ of households would earn $20\%$ of income. Any real distribution sags below that line, and the further it sags, the more unequal the distribution.

The Gini coefficient turns that sag into one number. Let $A$ be the area between the $45^{\circ}$ line and the Lorenz curve, and $B$ the area under the Lorenz curve down to the horizontal axis. Then

Because the whole triangle under the $45^{\circ}$ line has area $0.5$, this is the same as $G=2A$. The coefficient runs from $0$ (perfect equality, the curve is the line) to $1$ (one household owns everything).

Two limits the exam rewards. A Gini computed on market income ignores taxes and transfers, so it overstates inequality in actual living standards. And two very different distributions can produce the same Gini if their Lorenz curves cross — the single number hides where in the distribution the gap sits.

Progressive, proportional, regressive

A tax is classified by what happens to the average tax rate — tax paid divided by income — as income rises. Never classify by the dollar amount paid.

• Progressive: the average rate rises with income. Rich households pay a larger share of income. This is the structure that lowers the Gini coefficient.
• Proportional (flat): the average rate is constant. Everyone pays the same share, so the Gini is unchanged.
• Regressive: the average rate falls as income rises. Sales taxes and excise taxes on food or fuel are the standard examples, because low-income households spend a larger share of income on those goods.

The crucial exam distinction is marginal versus average rate. A progressive income tax has marginal rates that rise in brackets, but the average rate a household actually pays is always lower than its top marginal rate, since the earlier brackets are taxed at lower rates. Government redistribution combines progressive taxation with transfer payments — cash assistance, food benefits — and it is usually the transfers, not the tax code, that move the Lorenz curve most.